SE(3) Equivariance

The scalar output of an SE(3) equivariant model—such as a predicted binding energy or toxicity score—remains exactly identical regardless of how the input molecule is rotated or translated in 3D space. This eliminates the need for data augmentation with random orientations and guarantees that physically identical systems yield identical predictions. The model learns the intrinsic geometry rather than memorizing coordinate frames.

Unlike invariant models that discard all directional information, SE(3) equivariant networks maintain vector and tensor representations that rotate consistently with the input. When predicting atomic forces or dipole moments, the output vectors rotate exactly as the molecule rotates. This property is critical for:

• Force field prediction for molecular dynamics
• Transition dipole moments for spectroscopy
• Velocity updates in diffusion models

SE(3) equivariant architectures bake the symmetry of 3D space directly into their weight structure, dramatically reducing the number of independent parameters that must be learned. Rather than learning separate filters for every possible orientation, the model learns a single canonical filter that is transformed by the group action. This yields 10-100x improvements in sample efficiency compared to non-equivariant baselines on molecular property prediction tasks.

Modern SE(3) equivariant models like NequIP and MACE construct messages using Clebsch-Gordan tensor products of irreducible representations. This mathematical machinery allows the network to combine geometric features of different angular momenta (scalars, vectors, tensors) while strictly preserving equivariance. The resulting architectures can capture complex many-body interactions without the computational explosion of full tensor product spaces.

By construction, SE(3) equivariant models cannot violate fundamental physical symmetries. A predicted force field will always be curl-free and energy-conserving when derived as the gradient of a scalar potential. This hard constraint prevents unphysical artifacts such as:

• Spurious torque on isolated molecules
• Non-zero net forces at equilibrium geometries
• Violation of angular momentum conservation

SE(3) equivariant architectures naturally disentangle what an atom is from where it is located. Scalar features encode chemical identity and local bonding environment, while vector and tensor features encode directional relationships. This separation enables transfer learning across chemically distinct systems that share similar geometric motifs, and allows the model to reason about shape complementarity independently of atom types.

A computationally efficient architecture that achieves E(n) equivariance without requiring expensive higher-order tensor products. EGNNs operate directly on scalar features (like atom types) and vector features (like coordinates), updating them through message passing. This design makes them significantly faster than tensor-product-based methods while maintaining strict equivariance to rotations, translations, and reflections, making them ideal for large-scale molecular dynamics simulations.

A locally equivariant neural network layer that builds upon point clouds by learning to map geometric point features to higher-order tensor fields. It uses learned filters conditioned on relative positions to construct features that transform predictably under SE(3) transformations. This architecture forms the theoretical backbone for many modern equivariant models, enabling the construction of complex geometric representations that respect the symmetries of 3D space.

A transformer architecture that integrates SE(3)/E(3) equivariance using tensor products and equivariant attention mechanisms. It achieves state-of-the-art performance on 3D molecular property prediction by combining the global context modeling of transformers with the geometric rigor of equivariant operations. Key innovations include:

• Equivariant attention that preserves tensor order
• Depth-wise tensor products for efficient higher-order interactions
• Application to the OC20 catalyst benchmark

An E(3)-equivariant neural network interatomic potential that uses tensor products of irreducible representations to achieve data-efficient and highly accurate force and energy predictions. NequIP demonstrates that building equivariance directly into the model architecture yields superior sample efficiency compared to data augmentation. It achieves state-of-the-art accuracy on molecular dynamics benchmarks while requiring orders of magnitude less training data than invariant models.

A highly accurate equivariant message-passing interatomic potential that leverages many-body expansions via higher-order tensor products. MACE systematically constructs body-ordered representations, achieving state-of-the-art efficiency and accuracy by combining:

• Atomic Cluster Expansion (ACE) for systematic feature construction
• Equivariant message passing for capturing long-range interactions
• Higher-order tensor products for modeling complex angular dependencies This design enables linear scaling with the number of atoms while maintaining high fidelity.

A generative model that learns to reverse a noising process on 3D atomic coordinates while maintaining E(3) or SE(3) equivariance. EDMs generate stable molecular conformers by:

• Adding Gaussian noise to atomic coordinates in an equivariant manner
• Training a denoising network that respects rotational and translational symmetries
• Sampling new geometries by iteratively removing noise This approach ensures that generated structures are physically valid and independent of the chosen coordinate frame.